diff options
Diffstat (limited to 'vorlesungen/slides/4/galois/aufloesbarkeit.tex')
| -rw-r--r-- | vorlesungen/slides/4/galois/aufloesbarkeit.tex | 106 |
1 files changed, 106 insertions, 0 deletions
diff --git a/vorlesungen/slides/4/galois/aufloesbarkeit.tex b/vorlesungen/slides/4/galois/aufloesbarkeit.tex index 3215689..ef5902b 100644 --- a/vorlesungen/slides/4/galois/aufloesbarkeit.tex +++ b/vorlesungen/slides/4/galois/aufloesbarkeit.tex @@ -4,11 +4,117 @@ % (c) 2021 Prof Dr Andreas Müller, OST Ostschweizer Fachhochschule % \begin{frame}[t] +\setlength{\abovedisplayskip}{5pt} +\setlength{\belowdisplayskip}{5pt} \frametitle{Auflösbarkeit} +\vspace{-20pt} \begin{columns}[t,onlytextwidth] \begin{column}{0.48\textwidth} +\uncover<2->{% +\begin{block}{Radikalerweiterung} +Automorphismen $f\in \operatorname{Gal}(\Bbbk(\alpha)/\Bbbk)$ +einer Radikalerweiterung +\[ +\Bbbk \subset \Bbbk(\alpha) +\] +sind festgelegt durch Wahl von $f(\alpha)$. + +\begin{itemize} +\item<3-> Warum: Alle $f(\alpha^k)$ sind auch festgelegt +\item<4-> $f(\alpha)$ muss eine andere Nullstelle des Minimalpolynoms sein +\end{itemize} + +\end{block}} +\uncover<8->{% +\begin{block}{Irreduzibles Polynom $m(X)\in\mathbb{Q}[X]$} +$\mathbb{Q}\subset \Bbbk$, +$n$ verschiedene Nullstellen $\mathbb{C}$: +\[ +\uncover<9->{ +\operatorname{Gal}(\Bbbk/\mathbb{Q}) +\cong +S_n} +\uncover<10->{ +\quad +\text{auflösbar?}} +\] +\end{block}} \end{column} \begin{column}{0.48\textwidth} +\begin{block}{\uncover<5->{Galois-Gruppen}} +\begin{center} +\begin{tikzpicture}[>=latex,thick] +\def\s{1.2} + +\uncover<2->{ +\fill[color=blue!20] (-1.1,-0.3) rectangle (0.3,{5*\s+0.3}); +\node[color=blue] at (-0.7,{2.5*\s}) [rotate=90] {Radikalerweiterungen}; +} + +\node at (0,0) {$\mathbb{Q}$}; +\node at (0,{1*\s}) {$E_1$}; +\node at (0,{2*\s}) {$E_2$}; +\node at (0,{3*\s}) {$E_3$}; +\node at (0,{4*\s}) {$\vdots\mathstrut$}; +\node at (0,{5*\s}) {$\Bbbk$}; +\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{0*\s}) -- (0,{1*\s}); +\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{1*\s}) -- (0,{2*\s}); +\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{2*\s}) -- (0,{3*\s}); +\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{3*\s}) -- (0,{4*\s}); +\draw[shorten >= 0.3cm,shorten <= 0.3cm] (0,{4*\s}) -- (0,{5*\s}); + +\begin{scope}[xshift=0.5cm] +\uncover<7->{ +\fill[color=red!20] (0,{0*\s-0.3}) rectangle (4.8,{5*\s+0.3}); +\node[color=red] at (4.5,{2.5*\s}) [rotate=90] {Auflösung der Galois-Gruppe}; +} +\uncover<5->{ +\node at (0,{0*\s}) [right] {$\operatorname{Gal}(\Bbbk/\mathbb{Q})$}; +\node at (0,{1*\s}) [right] {$\operatorname{Gal}(\Bbbk/E_1)$}; +\node at (0,{2*\s}) [right] {$\operatorname{Gal}(\Bbbk/E_2)$}; +\node at (0,{3*\s}) [right] {$\operatorname{Gal}(\Bbbk/E_3)$}; +\node at (1,{4*\s}) {$\vdots\mathstrut$}; +\node at (0,{5*\s}) [right] {$\operatorname{Gal}(\Bbbk/\Bbbk)$}; +\node at (1,{0.5*\s}) {$\cap\mathstrut$}; +\node at (1,{1.5*\s}) {$\cap\mathstrut$}; +\node at (1,{2.5*\s}) {$\cap\mathstrut$}; +\node at (1,{3.5*\s}) {$\cap\mathstrut$}; +\node at (1,{4.5*\s}) {$\cap\mathstrut$}; +} + +\uncover<6->{ +\begin{scope}[xshift=2.5cm] +\node at (0,{0*\s}) {$G_n$}; +\node at (0,{1*\s}) {$G_{n-1}$}; +\node at (0,{2*\s}) {$G_{n-2}$}; +\node at (0,{3*\s}) {$G_{n-3}$}; +\node at (0,{5*\s}) {$G_0=\{e\}$}; +\node at (0,{0.5*\s}) {$\cap\mathstrut$}; +\node at (0,{1.5*\s}) {$\cap\mathstrut$}; +\node at (0,{2.5*\s}) {$\cap\mathstrut$}; +\node at (0,{3.5*\s}) {$\cap\mathstrut$}; +\node at (0,{4.5*\s}) {$\cap\mathstrut$}; +} + +\uncover<7->{ +\node[color=red] at (0.2,{0.5*\s+0.1}) [right] {\tiny $G_n/G_{n-1}$}; +\node[color=red] at (0.2,{0.5*\s-0.1}) [right] {\tiny abelsch}; + +\node[color=red] at (0.2,{1.5*\s+0.1}) [right] {\tiny $G_{n-1}/G_{n-2}$}; +\node[color=red] at (0.2,{1.5*\s-0.1}) [right] {\tiny abelsch}; + +\node[color=red] at (0.2,{2.5*\s+0.1}) [right] {\tiny $G_{n-2}/G_{n-3}$}; +\node[color=red] at (0.2,{2.5*\s-0.1}) [right] {\tiny abelsch}; +} + +\end{scope} +\end{scope} + + + +\end{tikzpicture} +\end{center} +\end{block} \end{column} \end{columns} \end{frame} |